Graphical models for combining multiple data sources Nicky Best Sylvia Richardson Chris Jackson Imperial College BIAS node with thanks to Peter Green

Graphical models for combining multiple data sources

Nicky BestSylvia Richardson

Chris Jackson

Imperial College BIAS node

with thanks to Peter Green

Outline

• Overview of graphical modelling• Case study 1: Water disinfection byproducts and

adverse birth outcomes – Modelling multiple sources of bias in observational

studies

• Case study 2: Socioeconomic factors and limiting long term illness– Combining individual and aggregate level data– Simulation study – Application to Census and Health Survey for England

Graphical modelling

Modelling

Inference

Mathematics

Algorithms

1. Mathematics

Modelling

Inference

Mathematics

Algorithms

• Key idea: conditional independence• X and Y are conditionally independent given Z if, knowing

Z, discovering Y tells you nothing more about XP(X | Y, Z) = P(X | Z)

Example: Mendelian inheritance

• Z = genotype of parents • X, Y = genotypes of 2 children• If we know the genotype of the parents, then the

children’s genotypes are conditionally independent

Z

X Y

Joint distributions and graphical models

Use ideas from graph theory to: • represent structure of a joint probability

distribution…..• …..by encoding conditional independencies

• Factorization thm:

Jt distribution P(V) = P(v | parents[v])

D

EB

CA

F

Where does the graph come from?

• Genetics– pedigree (family tree)

• Physical, biological, social systems– supposed causal effects

• Contingency tables– hypothesis tests on data

• Gaussian case– non-zeros in inverse covariance matrix

• Conditional independence provides mathematical basis for splitting up large system into smaller components

D

EB

CA

F

• Conditional independence provides mathematical basis for splitting up large system into smaller components

D

EB

C

D

E

F

CA

2. Modelling

• Graphical models provide framework for building probabilistic models for empirical data

Modelling

Inference

Mathematics

Algorithms

Building complex models

Key idea• understand complex system• through global model• built from small pieces

– comprehensible– each with only a few variables– modular

Example: Case study 1

• Epidemiological study of birth defects and mothers’ exposure to water disinfection byproducts

• Background– Chlorine added to tap water supply for disinfection– Reacts with natural organic matter in water to form

unwanted byproducts (including trihalomethanes, THMs)– Some evidence of adverse health effects (cancer, birth

defects) associated with exposure to high levels of THM– We are carrying out study in Great Britain using routine

data, to investigate risk of birth defects associated with exposure to different THM levels

Data sources

• National postcoded births register• National and local congenital anomalies registers• Routinely monitored THM concentrations in tap

water samples for each water supply zone within 14 different water company regions

• Census data – area level socioeconomic factors• Millenium cohort study (MCS) – individual level

outcomes and confounder data on sample of mothers

• Literature relating to factors affecting personal exposure (uptake factors, water consumption, etc.)

Model for combining data sources

yzk

[c]

[T]

pzk

yzi

pzi

czk

z

czi

THMzk[pers]

THMzt[tap]

THMztj[raw]

THMzi[pers]


yzk

[c]

[T]

pzk

yzi

pzi

czk

z

czi

THMzk[pers]

THMzt[tap]

THMztj[raw]

THMzi[pers]

Regression model for national data relating risk of

birth defects (pzk) to mother’s THM exposure

and other confounders (czk)


yzk

[c]

[T]

pzk

yzi

pzi

czk

z

czi

THMzk[pers]

THMzt[tap]

THMztj[raw]

THMzi[pers]

Regression model for MCS data relating risk of birth defects (pzi) to mother’s

THM exposure and other confounders (czi)


yzk

[c]

[T]

pzk

yzi

pzi

czk

z

czi

THMzk[pers]

THMzt[tap]

THMztj[raw]

THMzi[pers]

Missing data model to estimate confounders (czk)

for mothers in national data, using information on within area distribution of

confounders in MCS


yzk

[c]

[T]

pzk

yzi

pzi

czk

z

czi

THMzk[pers]

THMzt[tap]

THMztj[raw]

THMzi[pers]

Model to estimate true tap water THM concentration

from raw data


yzk

[c]

[T]

pzk

yzi

pzi

czk

z

czi

THMzk[pers]

THMzt[tap]

THMztj[raw]

THMzi[pers]

Model to predict personal exposure using estimated tap water THM level and

literature on distribution of factors affecting individual

uptake of THM

3. Inference

Modelling

Inference

Mathematics

Algorithms

Bayesian

… or non Bayesian

Bayesian Full Probability Modelling

• Graphical approach to building complex models lends itself naturally to Bayesian inferential process

• Graph defines joint probability distribution on all the ‘nodes’ in the model

• Condition on parts of graph that are observed (data)

• Update probabilities of remaining nodes using Bayes theorem

• Automatically propagates all sources of uncertainty

4. Algorithms

• Many algorithms, including MCMC, are able to exploit graphical structure

• MCMC: subgroups of variables updated randomly• Ensemble converges to equilibrium (e.g. posterior) dist.

Modelling

Inference

Mathematics

Algorithms

MCMC

?

?Updating - need only look at neighbours

Key idea exploited by WinBUGS software

Case study 2

• Socioeconomic factors affecting health• Background

– Interested in individual versus contextual effects of socioeconomic determinants of health

– Often investigated using multi-level studies (individuals within areas)

– Ecological studies also widely used in epidemiology and social sciences due to availability of small-area data

• investigate relationships at level of group, rather than individual• outcome and exposures are available as group-level summaries• usual aim is to transfer inference to individual level

Building the model

x[c]ik

yik

[b]

[c]

i

pikx[b]ik

Multilevel model for individual data

Building the model


yik ~ Bernoulli(pik), person k, area i

x[c]ik

yik

[b]

[c]

i

pikx[b]ik

Building the model



x[c]ik

yik

[b]

[c]

i

pikx[b]ik

log pik = i + [c] x[c]ik + [b] x[b]ik

Building the model



x[c]ik

yik

[b]

[c]

i

pikx[b]ik


i ~ Normal(0, )

Building the model



x[c]ik

yik

[b]

[c]

i

pikx[b]ik


i ~ Normal(0, )

Prior distributions on 2, [c], [b]

Building the model

X[c]i

Yi

[b]

[c]

i

i X[b]i

V[c]i

Ni

Ecological model

Building the model

X[c]i

Yi

[b]

[c]

i

i X[b]i

V[c]i

Ecological model

Yi ~ Binomial(i, Ni), area i

Ni

Building the model

X[c]i

Yi

[b]

i

i X[b]i

V[c]i

Ecological model


i = pik(x[b], x[c]) fi(x[b], x[c]) dx[b]dx[c]

Ni

[c]

Assuming x[b], x[c] independent, with

X[b]i = proportion exposed to ‘b’ in area

i and fi(x[c]) = Normal(X[c]i, V[c]i), then

i = q0i(1-X[b]i) + q1iX[b]i

where

q0i = marginal prob of disease for unexposed

= exp(i + [c]X[c]I + 2[c]V[c]i/2)

Building the model

X[c]i

Yi

[b]

i

i X[b]i

V[c]i

Ecological model


i = pik(x[b], x[c]) fi(x[b], x[c]) dx[c]dx[c]

Ni

[c]

Assuming x[b], x[c] independent, with

X[b]i = proportion exposed to ‘b’ in area

i and fi(x[c]) = Normal(X[c]i, V[c]i), then

i = q0i(1-X[b]i) + q1iX[b]i

where

q1i = marginal prob of disease for exposed

= exp(i + [b] + [c]X[c]I + 2[c]V[c]i/2)

Building the model

X[c]i

Yi

[b]

i

i X[b]i

V[c]i

Ecological model



Ni

[c]

Building the model

X[c]i

Yi

[b]

[c]

i

i X[b]i

V[c]i

Ecological model



Ni

i ~ Normal(0, )

Building the model

X[c]i

Yi

[b]

[c]

i

i X[b]i

V[c]i

Ecological model



Ni

i ~ Normal(0, )

Prior distributions on 2, [b], [c]

Combining individual and aggregate data

• Individual level survey data often lack power to inform about contextual and/or individual-level effects

• Even when correct (integrated) model used, ecological data often contain little information about some or all effects of interest

• Can we improve inference by combining both types of model / data?


X[c]i

Yi

[b]

[c]

i

i X[b]i

V[c]i

Ni

x[c]ik

yik

[b]

[c]

i

pikx[b]ik


Ecological model


X[c]i

Yi

[b]

[c]

i

i X[b]i

V[c]i

Ni

x[c]ik

yik

pikx[b]ik

Hierarchical Related Regression (HRR) model

Simulation Study

Posterior mean beta[b](log OR for binary cov.)

0.2 0.4 0.6 0.8 1.00.2 0.4 0.6 0.8 1.0Ecological+ sample of 10

Wider continuousexposure range

Wider binaryexposure range

Basic case,individual data only

Basic case

Posterior mean beta[c](log OR for cts cov.)

0.5 0.7 0.9 1.10.5 0.7 0.9 1.1

Simulation Study


0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0Ecological+ sample of 10

Correlation,modelled

Correlation,ignored

Misspecified distribution

Estimated within-area variance


0.5 0.7 0.9 1.10.5 0.7 0.9 1.1

Simulation Study


0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0Ecological+ sample of 10

Large ME, individualdata only

Large measurementerror

Small measurementerror


0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0

Comments

• Inference from aggregate data can be unbiased provided exposure contrasts between areas are high (and appropriate integrated model used)

• Combining aggregate data with small samples of individual data can reduce bias when exposure contrasts are low

• Combining individual and aggregate data can reduce MSE of estimated compared to individual data alone

• Individual data cannot help if individual-level model is misspecified

Application to LLTI

• Health outcome– Limiting Long Term Illness (LLTI) in men aged 40-59 yrs

living in London

• Exposures– ethnicity (white/non-white), income, area deprivation

• Data sources– Aggregate: 1991 Census aggregated to ward level – Individual: Health Survey for England (with ward identifier)

• 1-9 observations per ward (median 1.6)

Ward level data

Deprivation

Deprivation

Dep

rivat

ion

% non white

% non white% non white

Mean income

Mea

n in

com

e

Mea

n in

com

e

Pre

vale

nce

of

LLT

I

Pre

vale

nce

of

LLT

I

Pre

vale

nce

of

LLT

I

ResultsModel Non-white Log income Deprivation Between-

area variance

Individual -0.36

(-0.98, 0.23)

-0.55

(-0.80, -0.32)

-0.022

(-0.032, 0.074)

0.18

(0.052, 0.64)

Ecological 0.50

(0.27, 0.72)

-0.72

(-0.93, -0.51)

0.063

(0.054, 0.073)

0.19

(0.17, 0.21)

Combined 0.48

(0.23, 0.72)

-0.70

(-0.91, -0.50)

0.064

(0.054, 0.074)

0.19

(0.17, 0.22)

Combined (correlation modelled)

0.50

(0.24, 0.73)

-0.71

(-0.91, -0.51)

0.064

(0.054, 0.073)

0.19

(0.17, 0.21)

Concluding Remarks

• Graphical models are powerful and flexible tool for building realistic statistical models for complex problems– Applicable in many domains– Allow exploiting of subject matter knowledge– Allow formal combining of multiple data sources– Built on rigorous mathematics– Principled inferential methods

Thank you for your attention!

Documents

Graphical models for combining multiple data sources Nicky Best Sylvia Richardson Chris Jackson Imperial College BIAS node with thanks to Peter Green